# 1.6 Inverse functions  (Page 7/10)

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For the following exercises, use function composition to verify that $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ are inverse functions.

$f\left(x\right)=\sqrt[3]{x-1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)={x}^{3}+1$

$f\left(x\right)=-3x+5\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\frac{x-5}{-3}$

## Graphical

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

$f\left(x\right)=\sqrt{x}$

one-to-one

$f\left(x\right)=\sqrt[3]{3x+1}$

$f\left(x\right)=-5x+1$

one-to-one

$f\left(x\right)={x}^{3}-27$

For the following exercises, determine whether the graph represents a one-to-one function.

not one-to-one

For the following exercises, use the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ shown in [link] .

Find $\text{\hspace{0.17em}}f\left(0\right).$

$3$

Solve $\text{\hspace{0.17em}}f\left(x\right)=0.$

Find $\text{\hspace{0.17em}}{f}^{-1}\left(0\right).$

$2$

Solve $\text{\hspace{0.17em}}{f}^{-1}\left(x\right)=0.$

For the following exercises, use the graph of the one-to-one function shown in [link] .

Sketch the graph of $\text{\hspace{0.17em}}{f}^{-1}.\text{\hspace{0.17em}}$

Find

If the complete graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is shown, find the domain of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$

$\left[2,10\right]$

If the complete graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is shown, find the range of $\text{\hspace{0.17em}}f.$

## Numeric

For the following exercises, evaluate or solve, assuming that the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is one-to-one.

If $\text{\hspace{0.17em}}f\left(6\right)=7,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}\text{\hspace{0.17em}}{f}^{-1}\left(7\right).$

$6$

If $\text{\hspace{0.17em}}f\left(3\right)=2,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}{f}^{-1}\left(2\right).$

If $\text{\hspace{0.17em}}{f}^{-1}\left(-4\right)=-8,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}f\left(-8\right).$

$-4$

If $\text{\hspace{0.17em}}{f}^{-1}\left(-2\right)=-1,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}f\left(-1\right).$

For the following exercises, use the values listed in [link] to evaluate or solve.

 $x$ $f\left(x\right)$ 0 8 1 0 2 7 3 4 4 2 5 6 6 5 7 3 8 9 9 1

Find $\text{\hspace{0.17em}}f\left(1\right).$

$0$

Solve $\text{\hspace{0.17em}}f\left(x\right)=3.$

Find $\text{\hspace{0.17em}}{f}^{-1}\left(0\right).$

$\text{\hspace{0.17em}}1\text{\hspace{0.17em}}$

Solve $\text{\hspace{0.17em}}{f}^{-1}\left(x\right)=7.$

Use the tabular representation of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ in [link] to create a table for $\text{\hspace{0.17em}}{f}^{-1}\left(x\right).$

 $x$ 3 6 9 13 14 $f\left(x\right)$ 1 4 7 12 16
 $x$ 1 4 7 12 16 ${f}^{-1}\left(x\right)$ 3 6 9 13 14

## Technology

For the following exercises, find the inverse function. Then, graph the function and its inverse.

$f\left(x\right)=\frac{3}{x-2}$

$f\left(x\right)={x}^{3}-1$

${f}^{-1}\left(x\right)={\left(1+x\right)}^{1/3}$

Find the inverse function of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x-1}.\text{\hspace{0.17em}}$ Use a graphing utility to find its domain and range. Write the domain and range in interval notation.

## Real-world applications

To convert from $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ degrees Celsius to $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ degrees Fahrenheit, we use the formula $\text{\hspace{0.17em}}f\left(x\right)=\frac{9}{5}x+32.\text{\hspace{0.17em}}$ Find the inverse function, if it exists, and explain its meaning.

${f}^{-1}\left(x\right)=\frac{5}{9}\left(x-32\right).\text{\hspace{0.17em}}$ Given the Fahrenheit temperature, $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ this formula allows you to calculate the Celsius temperature.

The circumference $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ of a circle is a function of its radius given by $\text{\hspace{0.17em}}C\left(r\right)=2\pi r.\text{\hspace{0.17em}}$ Express the radius of a circle as a function of its circumference. Call this function $\text{\hspace{0.17em}}r\left(C\right).\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}r\left(36\pi \right)\text{\hspace{0.17em}}$ and interpret its meaning.

A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, $\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}$ in hours given by $\text{\hspace{0.17em}}d\left(t\right)=50t.\text{\hspace{0.17em}}$ Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function $\text{\hspace{0.17em}}t\left(d\right).\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}t\left(180\right)\text{\hspace{0.17em}}$ and interpret its meaning.

$t\left(d\right)=\frac{d}{50},\text{\hspace{0.17em}}$ $t\left(180\right)=\frac{180}{50}.\text{\hspace{0.17em}}$ The time for the car to travel 180 miles is 3.6 hours.

## Functions and Function Notation

For the following exercises, determine whether the relation is a function.

$\left\{\left(a,b\right),\left(c,d\right),\left(e,d\right)\right\}$

function

$\left\{\left(5,2\right),\left(6,1\right),\left(6,2\right),\left(4,8\right)\right\}$

${y}^{2}+4=x,\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ the independent variable and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ the dependent variable

not a function

Is the graph in [link] a function?

For the following exercises, evaluate the function at the indicated values: $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(-3\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(2\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(-a\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-f\left(a\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(a+h\right).$

$f\left(x\right)=-2{x}^{2}+3x$

$f\left(-3\right)=-27;$ $f\left(2\right)=-2;$ $f\left(-a\right)=-2{a}^{2}-3a;$
$-f\left(a\right)=2{a}^{2}-3a;$ $f\left(a+h\right)=-2{a}^{2}+3a-4ah+3h-2{h}^{2}$

$f\left(x\right)=2|3x-1|$

For the following exercises, determine whether the functions are one-to-one.

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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Sherica
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Sherica
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Tamia
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a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
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Kristine 2*2*2=8
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
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Samantha
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Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
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Cesar
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
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Prasenjit
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Azam
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Prasenjit
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Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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